A note on the interior and dimensions of typical sumsets
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- by Jian-Ci Xiao;
- Proc. Amer. Math. Soc. 151 (2023), 4799-4806
- DOI: https://doi.org/10.1090/proc/16500
- Published electronically: July 21, 2023
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Abstract:
Let $\mathcal {K}$ be the collection of all non-empty compact subsets of the unit cube $[0,1]^d$. Fixing any $F_\sigma$ set $F\subset [0,1]^d$ of Lebesgue measure zero and $0<c<1$, we show that for typical $K\in \mathcal {K}$ with $\mathcal {L}^d(K)\geq c$ (in the sense of Baire category), the sumset $K+F$ has empty interior. We also find the Hausdorff, lower and upper box and packing dimensions of $K+F$ for typical $K\in \mathcal {K}$.References
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Bibliographic Information
- Jian-Ci Xiao
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
- MR Author ID: 1427859
- ORCID: 0000-0002-1014-8725
- Email: jcxiao@math.cuhk.edu.hk
- Received by editor(s): November 13, 2022
- Received by editor(s) in revised form: March 15, 2023, and March 26, 2023
- Published electronically: July 21, 2023
- Additional Notes: The research of the author was partially supported by the General Research Funds (CUHK14301017, CUHK14301218) from the Hong Kong Research Grant Council.
- Communicated by: Nageswari Shanmugalingam
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 4799-4806
- MSC (2020): Primary 28A75; Secondary 28A80
- DOI: https://doi.org/10.1090/proc/16500
- MathSciNet review: 4634883